Topological properties

For properties which arise when viewed as metric space, eg: compactness, boundedness, connectedness etc.., see topology ref.

Properties of $R^{n, C^{n}$

See properties of k-cells in R in analysis of functions over fields ref.

Completeness of $R^{n, C^{n}$

Take vector seq $(x_{i}) \to x$ . If V is over R or C, $(x_{i}) \to x$ iff it is a Cauchy seq wrt norm: from equiv property over R and C.

Dimension of V

This is the maximal size of any linearly independent set of vectors in V.

Basis of vector space V

$T = {t_{i}} : \forall v \in V : v = \sum a_{i} t_{i}$ , such that $T$ is linearly independent. Often written as a matrix T, so that we can write $T a = v$ for any v in V. Any maximal set of independent vectors in V is a basis. All bases (eg T and T’) have the same size: otherwise, you would have a contradiction. So, $| T | = d i m (V)$ .

Orthonormal and standard basis

Orthonormal basis: $t_{i}^{T} t_{j} = 0, ⟨ t_{i}, t_{i} ⟩ = 1$ . Standard basis: see section on definiton of vectors.

Can get an orthogonal basis using QR making algorithms.

Subspaces

Aka Linear subspace. For subspaces associated with a linear operator, see linear algebra ref.

Membership conditions

A vector $v \neq 0$ is in a subspace S iff it is some linear combination of its basis Q; so $v = Q x$ .

This happens only if $\exists i : v^{T} q_{i} \neq 0$ : so v is not $⊥ Q$ wrt standard inner product.

Invariant subspace

S is an invariant subspace of A if $A S \subseteq S$ .